Exercises: Square roots

Complete the following table in your notebook relating the squared power to its corresponding exact square root. Remember that the square root of a number $a$ is another number $b$ such that $b^2 = a$.

Power ($b^2 = a$)	Square root ($\sqrt{a} = b$)
$3^2 = 9$	$\sqrt{9} = 3$
$7^2 = \dots$	$\dots = \dots$
$11^2 = \dots$	$\sqrt{121} = \dots$
$\dots = 1600$	$\sqrt{1600} = 40$
$15^2 = 225$	$\dots = \dots$

We want to buy a square rug for a gaming room. We know that the total surface area covered by the rug is $144 \text{ m}^2$.

Find mentally what the side length of the rug should be. Use the following diagram to help you visualize the problem.

Find mentally the following basic operations with exact square roots. Remember to follow the order of operations (first solve the roots, then add or subtract).

1. $\sqrt{25} + \sqrt{64}$

2. $\sqrt{100} - \sqrt{1}$

3. $3 \cdot \sqrt{4} + \sqrt{81}$

4. $\sqrt{144} : \sqrt{9}$

Not all roots result in an integer. When the result is not exact, we say that the root lies between two consecutive integers.

Without using a calculator, state between which two consecutive integers the following roots lie:

• Example: $\sqrt{10}$ is between 3 and 4 (because $3^2=9$ and $4^2=16$).

• $\sqrt{30}$

• $\sqrt{55}$

• $\sqrt{90}$

• $\sqrt{5}$

In mathematics, it is very important not to confuse the properties of operations. Check if the root of a sum is equal to the sum of the roots with the following example.

Calculate separately:

1. $\sqrt{64 + 36}$ (First add the numbers inside and then take the root).

2. $\sqrt{64} + \sqrt{36}$ (Calculate each root separately and then add the results).

Do they give the same result? Conclude whether it is true that $\sqrt{a+b} = \sqrt{a} + \sqrt{b}$.

You have a collection of 196 identical square tiles to pave a square inner courtyard. If you use all the tiles without breaking any:

1. How many tiles will you have to put on each side of the courtyard?

2. If each tile has a side length of $50 \text{ in}$, how many feet is the total side of the courtyard?

Calculate the value of the unknown side ($x$) of the following squares knowing their area. Notice that in case b, the number is not an integer; you can leave it indicated in root form or use a calculator to approximate.

a) Area = $49 \text{ cm}^2$

b) Area = $15 \text{ cm}^2$

c) Area = $121 \text{ m}^2$

Identify which of the following numbers are perfect squares (their square root is exact) and which are not.

Calculate the root of those that are perfect and explain why the others are not.

List of numbers: $25$, $40$, $81$, $100$, $200$, $144$.

Calculate mentally or by prime factorization the exact value of the following square roots. If the result is a fraction or a decimal, express it correctly.

1. a) $\sqrt{121}$
2. b) $\sqrt{400}$
3. c) $\sqrt{0,36}$
4. d) $\sqrt{\frac{64}{225}}$

Not all numbers have an exact square root. Without using a calculator, identify between which two consecutive integers the following roots lie.

For example: $\sqrt{10}$ is between 3 and 4 because $3^2=9$ and $4^2=16$.

1. a) $\sqrt{20}$
2. b) $\sqrt{55}$
3. c) $\sqrt{110}$
4. d) $\sqrt{95}$

We want to fence the perimeter of a square plot intended for a school garden. We know that the total surface area of the plot is $196 \ m^2$.

1. a) Calculate the side length of the plot.
2. b) If we want to put a fence around it, how many linear feet of fencing do we need to buy?

Simplify the following expressions with square roots by extracting factors from the radical and grouping like terms. Remember that to add or subtract roots, they must have the same radicand.

Hint: Factorize the numbers inside the root first. For example $\sqrt{8} = \sqrt{2^2 \cdot 2} = 2\sqrt{2}$.

1. a) $3\sqrt{2} + 5\sqrt{2} - \sqrt{8}$
2. b) $\sqrt{12} + \sqrt{75} - \sqrt{3}$

Computer and tablet screens are measured by their diagonal in inches, but we want to calculate the dimensions in inches of a rectangular tablet.

If the base of the screen measures $24 \ cm$ and the height measures $10 \ cm$, calculate the exact length of the diagonal using the Pythagorean theorem. Is it an integer or an irrational number?

The physical education teacher wants to organize the 60 students of 8th grade into a perfect square formation for an exhibition (the same number of rows as columns).

1. a) Is it possible to form a perfect square with the 60 students without any left over? Explain your answer by looking for nearby perfect squares.
2. b) How many students would be left over if we try to form the largest possible square?
3. c) How many more students would be needed to form the next largest square?

Solve the following simple equations where the unknown is related to a square root or a square. Remember that the inverse operation of a root is a power.

1. a) $\sqrt{x} = 9$
2. b) $x^2 = 169$ (State the two possible solutions)
3. c) $\sqrt{x + 1} = 5$

Calculate the numerical value of the following expressions respecting the order of operations. Remember to solve the contents inside the radicals first.

1. a) $\sqrt{100 - 36}$ (Attention: it is not the same as $\sqrt{100} - \sqrt{36}$)
2. b) $3 \cdot \sqrt{16} + \sqrt{9} \cdot 2$
3. c) $\frac{\sqrt{144}}{2} - \sqrt{4 \cdot 9}$

To represent irrational numbers exactly on the real line, we use the Pythagorean theorem. We know that the hypotenuse of a right triangle with legs of length 1 is $\sqrt{1^2+1^2} = \sqrt{2}$. Following this logic:

Explain in detail, step by step, how you would represent the number $\sqrt{29}$ on the number line using a ruler and compass. You must state which integers you would use as measurements for the legs of the auxiliary right triangle. Use the following visual diagram to support your reasoning.

Simplify the following numerical expression fully by operating with the radicals. To do this, you must decompose the radicands into prime factors and extract all possible factors outside the root before adding or subtracting.

$$ 3\sqrt{8} - 5\sqrt{18} + \frac{1}{2}\sqrt{200} + \sqrt{50} $$

Once simplified, state whether the result is a rational or irrational number.

In a logistics warehouse, they use perfect cube-shaped boxes to optimize space. We know that the internal diagonal of the box (the line that goes from a lower corner to the opposite upper corner, passing through the center of the box) measures $\sqrt{27}$ feet.

Calculate the length of the edge (the side) of this box.

Hint: Remember that you must apply the Pythagorean theorem twice or use the formula for the diagonal of a cube.

We have an algebraic expression involving roots and powers. Simplify the following expression by extracting all possible factors outside the root. Assume that $a$, $b$ and $c$ are positive numbers.

$$ \sqrt{\frac{16a^5 \cdot b^7}{81c^4}} $$

The final result must be presented as a fraction without radicals in the denominator.

The speed $v$ (in feet per second) at which a car can take a curve without skidding depends on the radius $r$ of the curve (in feet), the coefficient of friction $\mu$ (dimensionless) and gravity $g$ ($9,8 \text{ ft/s}^2$). The formula is:

$$ v = \sqrt{\mu \cdot g \cdot r} $$

If an engineer designs a highway exit with a radius $r = 150 \text{ ft}$ and the asphalt has a coefficient of friction $\mu = 0,6$:

1. Calculate the theoretical maximum speed.

2. If for safety it is recommended to travel at a speed that is half the theoretical maximum, what would that approximate safe speed be in mph? (Use $\sqrt{882} \approx 29,7$).

Estimate without using a calculator the value of $\sqrt{60}$ to one decimal place. To do this:

1. Find the two perfect squares closest to 60.

2. Interpolate linearly or use a reasoned "trial and improvement" method to approximate the first decimal place.

3. Once you have your estimate, calculate the absolute error made knowing that the approximate real value is $7,74$.

Calculate the exact value of the following combined operation, respecting the order of operations. Write down all the intermediate steps.

$$ \sqrt{144} : (-3) + \sqrt{5^2 - 3^2} - (\sqrt{3})^2 \cdot \sqrt{4} $$

We have two square plots. The area of plot A is $450 \text{ ft}^2$ and the area of plot B is half that of plot A.

1. Write the exact expression for the side of plot B using square roots.

2. Simplify this root by extracting factors.

3. If we wanted to fence plot B, how many feet of fencing would we need? Give the simplified result in terms of $\sqrt{2}$.